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Consider the one-form 1 (-y dx + x dy) w = (x² + y²)ª on U = R² \ {(0,0)} , and the smooth function o : V → U, with V = {(r, t) E R² |r > 0}, given by $(r, t) = (r cos(t), r sin(t)). Find the pullback one-form ø* w on V. d*w =| dr + dt Note that o defines the well known change of coordinates from Cartesian to polar coordinates on U, where U is the xy-plane with the origin removed. We are using the convention here that the angle t can take any values, so $(r, t) = ¢(r,t +2rn) for any integer n E Z. In other words, our map ø is not injective .

Consider the one-form 1 (-y dx + x dy) w = (x² + y²)ª on U = R² \ {(0,0)} , and the smooth function o : V → U, with V = {(r, t) E R² |r > 0}, given by $(r, t) = (r cos(t), r sin(t)). Find the pullback one-form ø* w on V. d*w =| dr + dt Note that o defines the well known change of coordinates from Cartesian to polar coordinates on U, where U is the xy-plane with the origin removed. We are using the convention here that the angle t can take any values, so $(r, t) = ¢(r,t +2rn) for any integer n E Z. In other words, our map ø is not injective .

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the one-form W = (x² + y²)4 1 (-y dx + x dy) on U = R² \ {(0, 0)} , and the smooth function : V → U, with V = {(r, t) E R² | r > 0}, given by $(r, t) = (r cos(t), r sin(t)). Find the pullback one-form ø*w on V. **w = dr + |dt Note that o defines the well known change of coordinates from Cartesian to polar coordinates on U, where U is the xy-plane with the origin removed. We are using the convention here that the angle t can take any values, so 4(r, t) = ¢(r,t+2rn) for any integer n E Z. In other words, our map o is not injective .
Transcribed Image Text:Consider the one-form W = (x² + y²)4 1 (-y dx + x dy) on U = R² \ {(0, 0)} , and the smooth function : V → U, with V = {(r, t) E R² | r > 0}, given by $(r, t) = (r cos(t), r sin(t)). Find the pullback one-form ø*w on V. **w = dr + |dt Note that o defines the well known change of coordinates from Cartesian to polar coordinates on U, where U is the xy-plane with the origin removed. We are using the convention here that the angle t can take any values, so 4(r, t) = ¢(r,t+2rn) for any integer n E Z. In other words, our map o is not injective .
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